Global well-posedness for the derivative nonlinear Schrödinger equation

Abstract

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac{1}{2}}$ with mass strictly less than $4\pi$ or general initial conditions in the weighted Sobolev space $H^{2,2}$. In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $H^{\frac{1}{2}}$ and that furthermore the $H^{\frac{1}{2}}$ norm of the solutions remains globally bounded in time. One should recall that for $H^s$, with $s<1/2$, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.

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